Singular Perturbations of First-Order Hyperbolic Systems with Stiff Source Terms

  title={Singular Perturbations of First-Order Hyperbolic Systems with Stiff Source Terms},
  author={W. A. Yong},
  journal={Journal of Differential Equations},
  • W. Yong
  • Published 10 June 1999
  • Mathematics
  • Journal of Differential Equations
Abstract This work develops a singular perturbation theory for initial-value problems of nonlinear first-order hyperbolic systems with stiff source terms in several space variables. It is observed that under reasonable assumptions, many equations of classical physics of that type admit a structural stability condition . This condition is equivalent to the well-known subcharacteristic condition for one-dimensional 2×2-systems and the well-known time-like condition for one-dimensional scalar… 
Hyperbolic systems with relaxation: characterization of stiff well-posedness and asymptotic expansions
The Cauchy problem for linear constant-coefficient hyperbolic systems ut + ∑j A(j)uxj = (1/δ)Bu + Cu in d space dimensions is analyzed. Here (1/δ)Bu is a large relaxation term, and we are mostly
Basic Aspects of Hyperbolic Relaxation Systems
This article presents a systematic consideration of hyperbolic systems of first-order partial differential equations with source terms divided by a small parameter e. Starting with von Neumann’s
Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations
A Chapman-Enskog-type asymptotic expansion is introduced and an effective system of equations describing the late-time/stiff relaxation singular limit is derived, and a new finite volume discretization is proposed which allows for a discrete version of the same effective asymPTotic system.
Convergence rate from systems of balance laws to isotropic parabolic systems, a periodic case
It is proved that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant
We consider multidimensional hyperbolic systems of conservation laws with relaxation, together with their associated limit systems. A strong stability condition for such asymptotics has been
Coupling conditions for linear hyperbolic relaxation systems in two-scales problems
A discontinuous Galerkin (DG) scheme for solving the interface problem with the derived coupling condition and the L 2 stability is proposed and validated.
Relaxation Limit and Initial-Layers for a Class of Hyperbolic-Parabolic Systems
We consider a class of hyperbolic-parabolic systems with small diffusion terms and stiff sources. Existence of solutions to the Cauchy problem with ill prepared initial data is established by using
A stiffly stable semi-discrete scheme for the characteristic linear hyperbolic relaxation with boundary
We study the stability of the semi-discrete central scheme for the linear damped wave equation with boundary. We exhibit a sufficient condition on the boundary to guarantee the uniform stability of


A Quasi-linear, Singular Perturbation Problem of Hyperbolic Type
Using matched asymptotic expansions, a formal approximation can be constructed for an initial value problem of singularly perturbed, hyperbolic type in two independent variables. Under a time-like
Singular Perturbations of First-Order Hyperbolic Systems
This work develops the singular perturbation theory for initial-value problems of nonlinear first-order hyperbolic systems in several space variables. The results can be applied to many physical
The relaxation schemes for systems of conservation laws in arbitrary space dimensions
A linear hyperbolic system is constructed with a stiff lower order term that approximates the original system with a small dissipative correction and can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally.
Hyperbolic conservation laws with stiff relaxation terms and entropy
We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are
Zero relaxation and dissipation limits for hyperbolic conservation laws
We are interested in hyperbolic systems of conservation laws with relaxation and dissipation, particularly the zero relaxation limit. Such a limit is of interest in several physical situations,
Convergence to equilibrium for the relaxation approximations of conservation laws
We study the Cauchy problem for 2 × 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the
Hyperbolic conservation laws with relaxation
The effect of relaxation is important in many physical situations. It is present in the kinetic theory of gases, elasticity with memory, gas flow with thermo-non-equilibrium, water waves, etc. The
On singular singularly-perturbed initial value problems
Consider the vector initial value problem with for a singular matrix F(t) of constant rank with stable eigenvalues and zero eigenvalues having simple elementary divisors. This paper shows how to
Discrete Velocity Models of the Boltzmann Equation: A Survey on the Mathematical ASPECTS of the Theory
Discrete velocity models of the Boltzmann equation are of considerable conceptual interest in the kinetic theory of gases, and, at the same time, a fascinating mathematical subject. The last decade